A mathematical treatise on periodic structures under travelling loads with an application to railway tracks

Abstract

The paper deals with the vertical dynamics of a railway track. In the system under consideration a single rail is modelled as a Timoshenko beam. The rails are coupled by means of periodically spaced sleepers which are modelled as rigid bodies with two degrees-of-freedom. The railway track forms a typical periodic system consisting of a number of identical flexible elements (cells) which are coupled in an identical way (by means of the sleepers). The solution method applied in the paper consists in the direct application of Floquet’s theorem to the equations of motion of the beam structure. Arranging the periodic boundary conditions for the whole infinite system makes it possible to reduce the analysis to one cell of the periodic structure. There are two modes of travelling waves propagating in the two-dimensional periodic structure. The first mode corresponds to the in-phase propagation of waves in the two rails. The second mode represents the case of a half-wave-length phase difference between the waves. The solution for the system under moving harmonic forces consists of the sum of these two modes.